Advanced Calculus Single Variable

B.3 Norms On Linear Maps

To begin with, the notion of a linear map is just a function which is linear. Such a function,
denoted by L, and mapping ℝn to ℝm is linear means

(∑m ) ∑m
L xivi = xiLvi
i=1 i=1

In other words, it distributes across additions and allows one to factor out scalars. Hopefully
this is familiar from linear algebra. If not, there is a short review of linear algebra in the
appendix.

Definition B.3.1We use the symbol ℒ

(ℝn,ℝm )

to denote the space of lineartransformations which map ℝnto ℝm. For L ∈ℒ

(ℝn, ℝm )

one can always consider it as anm × n matrix A as follows. Let

( )
A = Le1 Le2 ⋅⋅⋅ Len

where in the above Leiis the ithcolumn. Define the sum and scalar multiple of lineartransformations in the natural manner. That is, for L,M linear transformations and α,βscalars,

(αL + βM )(x) ≡ αL (x)+ βM (x )

Observation B.3.2With the above definition of sums and scalar multiples of lineartransformations, theresult of such a linear combination of linear transformations is itselflinear. Indeed, forx,yvectors and a,b scalars,

(αL + βM )(ax+ by) ≡ αL (ax + by)+ βM (ax + by )

= αaL (x )+ αbL(y)+ βaM (x)+ βbM (y)

= a(αL(x) +βM (x))+ b(αL(y)+ βM (y))
= a(αL +βM )(x)+ b(αL +βM )(y )

Also, a linear combination of linear transformations corresponds to the linear combination ofthe corresponding matrices in which addition is defined in the usual manner as addition ofcorresponding entries. To see this, note that if A is the matrix of L and B the matrix ofM,

(αL + βM )ei ≡ (αA + βB )ei = αAei + βBei

by the usual rules of matrix multiplication. Thus the ithcolumn of

(αA + βB )

is thelinear combination of the ithcolumns of A and B according to usual rules of matrixmultiplication.

Proposition B.3.3For L ∈ℒ

(ℝn,ℝm )

, the matrix defined above satisfies

n
Ax = Lx, x ∈ ℝ

and if any m × n matrix A does satisfy Ax = Lx, then A is given in the abovedefinition.

Furthermore, you should verify that you can replace ≤ 1 with = 1 in the definition.
Thus

||A || ≡ sup{||Ax||ℝm : ||x||ℝn = 1}.

It is necessary to verify that this norm is actually well defined.

Lemma B.3.5The operator norm is well defined. Let A ∈ℒ

(ℝn, ℝm )

.

Proof: We can use the matrix of the linear transformation with matrix multiplication
interchangeably with the linear transformation. This follows from the above considerations.
Suppose limk→∞vk = v in ℝn. Does it follow that Avk→ Av? This is indeed the case with
the usual Euclidean norm and therefore, it is also true with respect to any other norm by the
equivalence of norms (Theorem B.2.33). To see this,